Li, Is). Then Proton and Neutron Are

8.,4. The Nucleon Isosp'in 225

8.4 The Nucleon Isospin

Charge independence of nuclear forces leads to the introduction of a new conserved quantum number, isospin. As early as 1932, Heisenberg treated the neutron and the proton as two states of one particle, the nucleon l/.(6) The two states presumably have the same mass, but the electromagnetic interaction makes the masses slightly different. (The mass difference of the u andd quarks also contributes, but we neglect this effect here and throughout this chapter.) To describe the two states of the nucleon, an isospin space (internal charge space) is introduced, and the following analogy to the two spin states of a spin-] particle is made:

Spin- j Particle in Ordinary Space Nucleon in Isospin Space Orientation Up Up, proton Down Down, neutron

The two states of an ordinary spin- j particle are not treated as two particles but as two states of one particle. Similarly, the proton and the neutron are considered as the up and the down state of the nucleon. Formally, the situation is described by introducing a new quantity, i,sospi,n /.(z) 16" nucleon with isospin ] has 2I *L:2 possible orientations in isospin space. The three components of the isospin vector .i are denoted by It,,Iz, and 13. The value of /3 distinguishes, bg d,efini,tion between the proton and the neutron. 13 : ++ is the proton and,l.3 : -; is the neutron.(8) The most convenient way to write the value of .I and 13 for a given state is by using a Dirac ket: lI, Is). Then proton and neutron are

proton neutron l+,+1, | *,-t). (8.13)

The charge for the particle 11,13) is given by

Q: e(Is + ;). (8.14)

With the values of the third component of 13 given in Eq. (8.13), the proton has charge e, and the neutron charge 0. 6W. Heisenberg, Z. Phgsik 77,I (Lg32). [tanslated in D. M. Brink, Nuclear Forces, pergamon, Elmsford, N.Y., 1965]. 7To distinguish spin and isospin, we write isospin vectors with an arrow. 8In nuclear physics, isospin r_s sometimes called isobaric spin; it is often denoted by T,and the neutron is taken to have Is : i and the proton Is - -j, b""u,rrru there are more neutrons than protons in stable nuclei and 13(73) is then positive for thlse cases. 226 Angular Momentum and" Isospin

8.5 Isospin Invariance

gained What have we with the introduction of isospin? So far, very little. Formally, ( the neutron and the proton can be described as two states of one particle. New 1 aspects and new results appear when charge independence is introduced and when ( isospin is generalized to all hadrons. ( Charge independence states forces that the hadronic do not distinguish between t the proton and the neutron. As long as only the hadronic interaction is present, ( the isospin vector i point in any direction. In other words, there exists rota- "un tional invariance in isospin space; the system is invariant under rotations about any direction. As in Eq. (8.10), this fact is expressed by

lu1, i1 : g. (8.15) \ With only H7 present, the 2I * 1 states with different vaiues of 13 are degenerate; l they have the same energy (mass). Said simply, with only the hadronic interaction t present, neutron and proton would have the same mass. The electromagnetic interaction (and the up-down quark mass difference) destroy the isotropy of isospin space; it breaks the symmetry, and, as in Eq. (8.11), it gives

lHn + H.^, il + o. (8.16)

However, we know from Section 7.1 that the electric charge is always conserved, even in the presence of H.rn;

lHn+ H.*,8] :0. (8.17)

Q is the operator corresponding to the electric charge q; it is connected to .I3 by Eq. (8.1a), Q --e(Is+ ]). Introducing Q into the commutator, Eq. (8.17), gives lHn+H.*,1e] :0. (8.18)

The third component of isospin is conserved even in the presence of the electromagnetic interaction. The analogy to the magnetic field case is evident; Eq. (8.18) is the isospin equivalent of Eq. (8.12). It was pointed out in Section 8.4 that charge independence holds not only for nucleons but for all hadrons. Before generalizing the isospin concept to all hadrons and exploring the consequences of such an assumption, a few preliminary remarks trc are in order concerning isospin space. We stress that l is a vector in isospin space, at not in ordinary space. The direction in isospin space has nothing to do with any tl direction in ordinary space, and the value of the operator i or Is in isospin space has nothing to do with ordinary space. So far, we have related only the third component of i to an observable, the electric charge q (Eq. (8.14)). What is the physical significance of 11 and 12? These two quantities cannot be connected directly T to a physically measurable quantity. The reason is nature: in the laboratorv, two p 8.5. Isospin Inuariance magnetic fields can be set up. The first can point in the z direction, and the second in the r direction. The effect of such a.combination on the spin of the particle can be computed, and the measurement along any direction is meaningful (within the limits of the uncertainty relations). The electromagnetic field in the isospin space, however, cannot be switched on and off. The charge is always related to one component of 1-, and this component is traditionally taken to be ,I3. Renaming the components and connecting the charge, for instance, to 12 does not change the situation. We now assume the general existence of an isospin space, with its third component connected to the charge of the particle by a linear relation of the form

Q:alslb' (8.1e)

With such a relationship, conservation of the electric charge implies conservation of Iz. Iz is therefore a good quantum number, even in the presence of the electromagnetic interaction. The unitary operator for a rotation in isospin space by an angle r,.r about the direction d is

Ua(r): exP(-z ua' i), (8.20)

where / is the Hermitian generator associated with the unitary operator (J, and we expect ito b. an observable. As in the case of the anguiar momentum operator J,, the arguments follow the general steps outlined in Section 7.1. To study the physical properties of 1-, we assume first that only the hadronic interaction is present. Then the electric charge is zero for all systems, and Eq. (8.19) does not determine the direction of .I3. Charge independence thus implies that a hadronic system without electromagnetic interaction is invariant under any rotation in isospin space. We know from Section 7.1, Eq. (7.9), that U then commutes with f11:

lHn,Ua(r)l :0. (8.21)

As in Eq. (7.15), conservation of isospin follows immediately,

lu1, i1 : g.

Charge independence of the hadronic forces leads to conservation of isospin. In the case of the ordinary angular momentum, the commutation relations for J follow from the unitary operator (8.7) by straightforward algebraic steps. No further assumptions are involved. The same argument can be applied to Ua(u,'), and the three components of the isospin vector must satisfy the commutation relations

llt,Izl : 'ilz, llr,Is) : i,11, [Is,1t] -- ilz. (8.22)

The eigenvalues and eigenfunctions of the isospin operators do not have to be computed because they are analogous to the corresponding quantities for ordinary spin. Angular Momentum and Isospin

The steps from Eq. (5.6) to Eqs. (5.7) and (5.8) are independent of the physical interpretation of the operators. All results for ordinary angular momentum can be taken over. In particular, 12 and 13 obey the eigenvalue equations

I3olI,1e) : I(I 1)l/,13) (8.23) + t Is,ooll ,/e) : hll , hr. (8.24) 0

ru Here /jo and .I3,oo orr the left-hand side are operators, and 1 and ,I3 on the right- s hand side are quantum numbers. The symbol 11, ,I3) denotes the eigenfunction tf;1,7". a (In a situation where no confusion can arise, the subscripts "op" will be omitted.) b The allowed values of ,I are the same as for J, Eq. (5.9), and they are ft (8.25) et n For each value of. I,Is can assume the 2I + l values from -I to I. tt In the following sections, the results expressed by Eqs. (8.22)-(8.25) will be ol applied to nuclei and to particles. It will turn out that isospin is essential for pl understanding and classifying subatomic particles. Yt o We have noted above that the components.Il and 12 are not directly connected st, to observables. However, the linear combinations iu

CU Ia -- It I i,Iz (8.26) have a physical meaning. Applied to a state l/,13),[ raises and 1- lowers the value of ,I3 by one unit: ap r+lr,,1s) : [(r+1r)(/rIa * r)]t/rv,,h+ 1). (8.27) ler rq Equation (8.27) can be derived with the help of Eqs. (8.22) to (S.Za;.tol . II c€rl 8.6 Isospin of Particles po' sel, The isospin concept was first applied to nuclei, but it is easier to see its salient features in connection with particles. As stated in previous the section, isospin is ral presumably a good quantum number as Iong as only the hadronic interaction is pio present. The electromagnetic interaction destroys the isotropy of isospin space, just as a magnetic field destroys the isotropy of ordinary space. Isospin and its mani- festations should consequently appear most clearly in situations where the electromagnetic interaction is small. For nuclei, the total electric charge number Z canbe sho as high as 100, whereas for particles it is usually 0 or 1. Isospin should therefore be gul a better and more easily recognized quantum number in particle physics. onll If isospin is an observable that is realized in nature, then Eqs. (8.15) and (8.23)- exis (8.25) predict the following characteristics: The quantum number 1 can take on Section XIII.I. -10 Isospi.n of Particles 229 the valuer 0,+ ,,1,3,.... For a given particle, 1 is an immutable property. In the absence of the electromagnetic interaction, a particle with isospin I is (21+ 1)-fold degenerate, and the 2I i I subparticles all have the same mass.

Since H7, and -f commute, all subparticles have Hr^ off Hr on the same hadronic properties and are differentiated only by the value of .I3. The electromagnetic interaction partially or completely lifts the degeneracy, as shown in Fig. 8.2, and it thus gives rise to the isospin analog of the Zeeman effect. The 2I * 1 subparticles belonging to a given state with isospin 1 are said to -l form an 'isospi,n multi,plet. The electric charge of each member is related to ,I3 by Eq. (8.19). Quan- t tum numbers that are conserved by the electromag- I, netic interaction are unaffected by the switching on (21+l)-fold degenerate of H"*. Since most quantum numbers have this property, the members of an isospin multiplet have Figure 8.2: A particle with isospin I is (21* l)-fold degen- very nearly identical properties; they have, for in- erate in the absence of the elec- stance, the same spin, baryon number, hypercharge, tromagnetic interaction. H.rn lifts the and intrinsic parity. (Intrinsic parity will be dis- degeneracy, and the re- sulting subparticles are labeled cussed in Section 9.2.) bv Is.

The different members of an isospin multiplet are in essence the same particle appearing with different orientations in isospin space, just as the various Zeeman levels are states of the same particle with different orientations of its spin with respect to the applied magnetic field. The determination of the quantum number / for a given state is straightforward if all subparticles belonging to the multiplet can be found: Their number is 2I * 1 and thus yields 1. Sometimes counting is not possible, and it is then necessary to resort to other approaches, such as the use of selection rules. The arguments given so far can be applied most easily to the pion. The possible values of the isospin of the pion can be found by Iooking at Fig. 5.19: If virtual pions are exchanged between nucleons, the basic Yukawa reaction

ly' -----+ N' + n should conserve isospin. Nucleons have isospin f ; isospins add vectorially like angular momenta, and the pion consequently must have isospin 0 or 1. If 1 were 0, only one pion would exist. The assignment I : l, on the other hand, implies the existence of three pions.(to) Indeed, three and only three hadrons with mass of

10N. Kemmer, Proc. Cambrid,ge Phit. Soc. g4,JS4 (1g3S). Angular Momentum and Isospi,n about 140 MeV/ c2 are known, and the three form an 'isouector with the assignment

( +I r* , m : 139.569 MeY f c2 , Is:{ 0 To, m:1.34.964MeYfc2, [ -t r , m:739.569MeYfc2.

The charge is connected to -I3 by the relation

Q -- elz, (8.28) which is a special case of Eq. (8.19). The pion shows particularly clearly that the properties in ordinary and in isospin space are not related because it is a vector in isospin space but a scalar (spin 0) in ordinary space. { In the ordinary Zeeman effect, it is easy to demonstrate that the various sublevels i are members of one Zeeman multiplet: if the applied rnagnetic field is reduced to t zero) they coalesce into one degenerate level. This method cannot be applied to an isospin multiplet because the electromagnetic interaction cannot be switched I off. It is necessary to resort to calculations to show that the observed splitting n can be blamed solely on H"n. Comparison of the pion and the nucleon shows l that the problem is not straight forward: the proton is iighter than the neutron, ir whereas the charged pions are heavier than the neutral one. Nevertheless, the I[ computations performed up to the present time account for the mass splitting by !E the electromagnetic interaction and the mass difference between the up and down ffi quarks.(11) c After having spent considerable time on the isospin of the pion, the other hadrons rel n can be discussed more concisely. 6( The kaon appears in two particle and two antiparticle states. The assignment pr t : L is in agreement with all known facts. The assignment of 1 to hyperons is also straightforward. It is assumed that ffi hyperons with approximately equal masses form isospin multiplets. The lambda wr occurs alone, and it is a singlet. The sigma shows three charge states, and it is an pe isovector. The cascade particle is a doublet, and the omega is a singlet. & The hadrons encountered so far can all be characterized by . set of additive firh quantum numbers, A, e, Y, and 13. For pions, charge and 13 are connected by he, Eq. (8.28). Gell-Mann and Nishijima showed how this relation can be generalized fuu to apply also to strange particles. They assumed charge and .I3 to be connected by 0tu! a linear relation as in Eq. (8.19). The constant a in Eq. (8.19) is determined from ilmi Eq. (8.28) as e. To find the constant b, we note that -I3 ranges from -1 to *1. The msf average charge of a multiplet is therefore equal to b: r-t ftw (q) : u. 1lSee e.g., A. De Rrijula, H. Georgi, and S. L. Glashow, Phys. Reu. Dt2,La7 Q975); N. Isgur and G. Karl, Phys. Reu. D 20,1191 (1979); J. Gaisser and H. Leutwyler, Phys. Repts. 87,77 (1932); E.M. Henley and G.A. Miller, Nucl. Phys. A5L8,207 (1990). 8.6. Isosp'in of Particles 2JI

The average charge of a multiplet has already been determined in Eq. (7.49):

h) : |"v. (8.2e)

Only particles with zero hypercharge have the center of charge of the multiplet at Q : 0; for all others, it is dispiaced. Consequently the generalization of Eqs. (8.14) and (8.28) is

e : e(Iz + +Y) : e(Iz + +A+ +s). (8.30)

This equation is called the Gell-Mann-Nishijima relation. If q is considered to be an operator, it can be said that the electric charge operator is composed of an isoscalar G"V) and the third component of an isovector ("Ir). For particles with charm, bottom, or top quantum numbers, Y in Eq. (8.30) is replaced by Ygen, Eq. (7.50).

The Gell-Mann-Nishijima relation can be visualized in a Y versus qf e diagram, shown in Fig. 8.3. A few isospin multiplets are plotted. The

multipletswithY + 0are lqle) = di,splaced: Their center of / charge is not at zero but, as expressed by Eq. (8.29), at 7oV 2-t ' The considerations in the present section have shown qle that isospin is a useful quantum number in particle phvsics. The value of 1 for a given particle determines the number of subparticles O Center of charge belonging to this particular isospin multiplet. The third component, Is, is conserved in hadronic and electromagnetic interactions, whereas Figure 8.3: Isospin multiplets with Y 0 are displaced: i i, only by the f "onrerved Their center of charge (average charge) is at jeY. A few hadronic force. representative multiplets are shown, but many more exist.

In the following section we shail demonstrate that isospin is also a valuable concept in nuclear physics.